CAT (QA) PYQs

Let $D$ and $E$ be points on sides $A B$ and $A C$, respectively, of a triangle $A B C$, such that $A D: B D=2: 1$ and $A E: C E=2: 3$. If the area of the triangle $A D E$ is $8 \mathrm{sq} \mathrm{cm}$, then the area of the triangle $ABC$ , in sq cm , is

The length of each side of an equilateral triangle ABC is $3 \mathrm{~cm}$. Let D be a point on BC such that the area of triangle ADC is half the area of triangle $A B D$. Then the length of $A D$, in cm , is

The coordinates of the three vertices of a triangle are: $(1, 2)$, $(7, 2)$, and $(1, 10)$. Then the radius of the in circle of the triangle is